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Görtler vortices in growing boundary layers: The leading edge receptivity problem, linear growth and the nonlinear breakdown stage

  • Philip Hall (a1)


Görtler vortices are thought to be the cause of transition in many fluid flows of practical importance. In this paper a review of the different stages of vortex growth is given. In the linear regime nonparallel effects completely govern this growth and parallel flow theories do not capture the essential features of the development of the vortices. A detailed comparison between the parallel and nonparallel theories is given and it is shown that at small vortex wavelengths the parallel flow theories have some validity; otherwise nonparallel effects are dominant. New results for the receptivity problem for Gortler vortices are given; in particular vortices induced by free-stream perturbations impinging on the leading edge of the wall are considered. It is found that the most dangerous mode of this type can be isolated and its neutral curve is determined. This curve agrees very closely with the available experimental data. A discussion of the different regimes of growth of nonlinear vortices is also given. Again it is shown that, unless the vortex wavelength is small, nonparallel effects are dominant. Some new results for nonlinear vortices of O(l) wavelengths are given and compared with experimental observations. The agreement between theory and experiment is shown to be excellent up to the point where unsteady effects become important. For small wavelength vortices the nonlinear regime is of particular interest since a strongly nonlinear theory can be developed there. Here the vortices can be large enough to drive the mean state which then adjusts itself to make all modes neutral. The breakdown of this nonlinear state into a three-dimensional time dependent flow is also discussed.



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Aihara, Y. (1961). Stability of compressible boundary-layers along a curved wall under Görtler type disturbances. Aero. Res. Inst. Tokyo. Rep., 362, 31.
Aihara, Y. (1962). Transition in incompressible boundary-layer along a concave wall. Bull. Aero. Res. lnst. Tokyo Univ., 3.
Aihara, Y. (1976). Nonlinear analysis of Görtler vortices. Phys. Fluids, 19, 1655.
Aihara, Y. and Kohama, H. (1981). Secondary instability of Gortler vortices: Formation of periodic three-dimensional coherent structure. Trans. Japan Soc. Aero. Sci., 24, 78.
Allison, D. O. and Dagenhart, J. R. (1987). Two experimental supercritical laminar flow control swept-wings. NASA TM 87073.
Baskaran, V. and Bradshaw, P. (1988). Decay of spanwise inhomogeneities in a three-dimensional turbulent boundary layer over an infinite swept concave wall. Experiments in Fluids, 6, 487.
Bassom, A. P. and Hall, P. (1988). On the generation of mean flows by the interaction of Görtler vortices and Tollmien-Schlichting waves in curved channel flows. ICASE Report 88-43 and to appear in Studies in Applied Maths.
Bippes, H. (1972). Experimental study of the laminar-turbulent transition of a concave wall in a parallel-flow. NASA TM 75243.
Bippes, H. and Görtler, H. (1972). Dreidimensionale Störungen in der Grenzschict an einer konkaven wand. Acta Mechanica, 14, 251.
Bouthier, M. (1973). Stabilité linéaire des écoulements presque parallèles: Partie II. La Couche Limite de Blasius. J. Mecanique, 12, 75.
Davey, A. (1962). The growth of Taylor vortices in flow between rotating cylinders. J. Fluid Mech., 14, 336.
Davey, A., DiPrima, R. C. and Stuart, J. T. (1968). On the instability of Taylor vortices. J. Fluid Mech., 31, 17.
Drazin, P. G. and Reid, W. H. (1979). Hydrodynamic Instability (CUP, Exercise 3.8).
El-Hady, N. and Verma, A. K. (1981). Growth of Görtler vortices in compressible boundary-layers along curved surfaces. A1AA Paper 811278.
Finnis, M. V. and Brown, A. (1988). Stability of a laminar boundary-layer flowing along a concave wall. ASME Paper, 88-GT-40.
Floryan, J. M. (1986). Görtler instability of boundary-layers over concave and convex walls. Phys. Fluids, 29, 2380.
Floryan, J. M. and Saric, W. S. (1979). Stability of Görtler vortices in boundary-layers. AIAA J., 20, 316.
Gaster, M. (1962). A note on the relationship between temporally-increasing and spatially-increasing disturbances in hydrodynamic instability. J. Fluid Mech., 14, 222.
Gaster, M. (1974). On the effects of boundary-layer growth on flow stability. J. Fluid Mech., 66, 465.
Görtler, H. (1940). Über eine dreidimensionale instabilität laminarer Grenzschichten an Konkaven wänden. Ges. D. Wiss. Göttingen, Nachr., 1, number 2.
Görtler, H. and Witting, H. (1958). Theorie der secundaren Instabilitat der laminaren Grenzschichten. Proc. IUTAM Symp. on boundary-layer research, Freiburg 1957, 110.
Gregory, N., Stuart, J. T. and Walker, W. S. (1955). On the stability of three-dimensional boundarylayers with application to the flow due to a rotating disc. Phil. Trans. Roy. Soc., A248, 155.
Gregory, N. and Walker, W. S. (1956). The effect on transition of isolated surface excrescences in the boundary-layer. ARC Rep., M2779.
Hall, P. (1982a). Taylor-Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech., 124, 475.
Hall, P. (1982b). On the nonlinear evolution of Görtler vortices in growing boundary layers. J. Inst. Maths. and Applications, 29, 173.
Hall, P. (1983). The linear development of Görtler vortices in growing boundary-layers. J. Fluid Mech., 126, 357.
Hall, P. (1985). The Görtler vortex instability mechanism in three-dimensional boundary layers. Proc. R. Soc., A399, 135.
Hall, P. (1986). An asymptotic investigation of the stationary modes of instability of the boundary-layer on a rotating disk. Proc. Roy. Soc., A406, 93.
Hall, P. (1988). The nonlinear development of Görtler vortices in growing boundary-layers. J. Fluid Mech., 193, 243.
Hall, P. and Bennett, J. (1986). Taylor-Görtler instabilities of Tollmien-Schlichting waves and other flows governed by the interactive boundary-layer equations. J. Fluid Mech., 171, 441.
Hall, P. and Fu, Y. (1989a). The Görtler vortex instability mechanism in hypersonic boundary-layers. J. Theor. Comp. Fluid Mech., 1, 125.
Hall, P. and Fu, Y. (1989b). The Görtler vortex instability in compressible boundary-layers. To appear in Proc. IUTAM Symp. Instability and Transition, Toulouse.
Hall, P. and Lakin, W. D. (1988). The fully nonlinear development of Görtler vortices. Proc. Roy. Soc., A415, 421.
Hall, P. and Malik, M. (1989). The growth of Görtler vortices in compressible boundary-layers. J. Eng. Math., 23, 239.
Hall, P. and Seddougui, S. O. (1989). On the onset of three-dimensionality and time-dependence in the Görtler problem. J. Fluid Meek, 204, 405.
Hall, P. and Smith, F. T. (1988). The nonlinear interaction of Tollmien-Schlichting waves and Taylor-Görtler vortices in curved channel flows. Proc. Roy. Soc., A417, 255.
Hall, P. and Smith, F. T. (1989a). Nonlinear Tollmien-Schlichting/vortex interaction in boundary-layers. Eur. J. Mech. B/Fluids, 8, 179.
Hall, P. and Smith, F. T. (1989b). On strongly nonlinear vortex/wave interactions in boundary-layer transition. To appear in J. Fluid Mech.
Hammerlin, G. (1955). Über das Eigenwertproblem der dredimensionalen Instabilität laminare Grenzschichten ab konkaven Wänden. J. Rat. Mech. Anal., 4, 279.
Hammerlin, G. (1956). Zur theorie der dreidimensionalen instabilität laminarer Grenzschichten. ZAMP, 7, 156.
Harvey, W. D. and Pride, J. D. (1982). The NASA Langley laminar flow control airfoil experiment. AIAA paper 820567.
Herbert, T. (1976). On the stability of the boundary-layer on a concave wall. Arch. Mech. Stos., 28, 1039.
Horseman, N. J. (1990). Some centrifugal instabilities in viscous flows. Exeter University PhD thesis, to be submitted.
Jallade, S. (1989). Effect of streamwise curvature on Görtler vortices. To appear in Proceedings of ICASE Workshop on Instability and Transition (Springer).
Kahawita, R. A. and Meroney, R. N. (1977). The influence of heating on the stability of laminar boundary layers along curved walls. J. Appl. Mech., 44, 11.
Kalburgi, V., Mangalam, S. M., Dagenhart, J. R. and Tiwari, S. N. (1987). Görtler instability on an airfoil, NASA Symposium on Natural Laminar Flow and Laminar Flow Control Research. To appear as a NASA TM.
Kalburgi, V., Mangalam, S. M. and Dagenhart, J. R. (1988). A comparative study of theoretical methods on Görtler instability. AIAA Paper, 880407.
Kieffer, S. W. and Sturtevant, B. (1988). Erosional furrows formed during the lateral blast at Mount St. Helens, May 18, 1980. J. Geophys. Res., 93, 14793.
Kobayashi, R. and Kohama, K. (1977). Taylor-Görtler instability of compressible boundary-layers. AIAA J., 15, 1723.
Kohama, Y. (1987). Three-dimensional boundary-layer transition on a concave-convex wall. Proc. Bangalore IUTAM Symp. Turbulence Management and Relaminarization, ed. Liepmann, H. and Narisimha, R..
Kottke, V. (1986). Taylor Görtler vortices and their effect on heat transfer. Proc. 8th Heat Transfer Conf. San Francisco, 1139.
Liepmann, H. W. (1943). Investigations on laminar boundary-layer stability and transition on curved boundaries. NACA Wartime Report, W107.
Liepmann, H. W. (1945). Investigation of boundary-layer transition on concave walls. NACA Wartime Report, W87.
Malik, M., Zang, T. A. and Hussaini, M. Y. (1985). A spectral collocation method for the Navier-Stokes equations. J. Comput. Phys., 61, 64.
Malkus, W. (1956). Outline of a theory of turbulent shear flow. J. Fluid Mech., 1, 521.
Mangalam, S. M., Dagenhart, J. R., Hepner, T. E. and Meyers, J. F. (1985). The Görtler instability on an airfoil. AIAA Paper, 850491.
Mangalam, S. M., Dagenhart, J. R. and Meyers, J. F. (1987). Experimental Studies on Görtler vortices. NASA Symposium on Natural Laminar Flow and Laminar Flow Control Research. To appear as a NASA TM.
Meksyn, D. (1950). Stability of viscous flow over concave cylindrical surfaces. Proc. Roy. Soc., A203, 116.
Park, D. and Huerre, P. (1988). On the convective nature of the Görtler instability. Bull. Amer. Physical Soc., 33, 2252.
Peerhossaini, H. and Wesfreid, J. E. (1988). On the inner structure of streamwise Görtler rolls. Intl. J. Heat Fluid Flow, 9, 12.
Pfenninger, W., Reed, H. L. and Dagenhart, J. R. (1980). Design considerations of advanced supercritical low-drag suction airfoils. Viscous Flow Drag Reduction, ed. Hough, G. R., Progress in Astro, and Aero., 72, 249.
Rayleigh, Lord (1916). On the dynamics of revolving fluids. Proc. Roy. Soc., A93, 148.
Sabry, A. and Liu, J. C. T. (1987). Nonlinear development of Görtler vortices and the generation of high shear layers in the boundary-layer. Proc. Symp. in Honour of C. C. Lin (World Scientific Books).
Seminara, G. and Hall, P. (1975). Centrifugal instability of a Stokes layer: linear theory. Proc. Roy. Soc., A350, 299.
Smith, A. M. O. (1955). On the growth of Taylor-Görtler vortices along highly concave walls. Quart. Appl. Math., 13, 233.
Smith, F. T. (1979). On the nonparallel stability of the Blasius boundary-layer. Proc. Roy. Soc., A366, 91.
Spall, R. E. and Malik, M. (1989). Görtler vortices in supersonic and hypersonic boundary-layers. Phys. Fluids, A1, 1822.
Swearingen, J. D. and Blackwelder, R. F. (1983). Parameters controlling the spacing of streamwise vortices on concave walls. AIAA Paper, 830380.
Swearingen, J. D. and Blackwelder, R. F. (1987). The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech., 182, 255.
Tani, I. (1962). Prediction of longitudinal vortices in the boundary-layer along a curved wall. J. Geophys. Research, 67, 3075.
Tani, I. and Sakagami, J. (1962). Boundary-layer instability at subsonic speeds. Proc. ICAS. III Congress Stockholm, 1962, 391.
Taylor, G. I. (1923). Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc., A223, 289.
Tobak, M. (1971). On local Görtler instability. ZAMP, 22, 130.
Wadey, P. (1989). Görtler vortex instabilities of incompressible and compressible boundary-layers. Exeter University Ph.D. thesis.
Winoto, S. H. and Crane, R. I. (1980). Vortex structure in laminar boundary-layers on a concave wall. Int. J. Heat and Fluid Flow, 2, 221.
Witting, H. (1958). Über den einfluss der Stromlinienkrümmung auf die Stabilität laminarer Stromungen. Arch. Rat. Mech. Anal., 2, 243.
Wortmann, F. X. (1964a). Experimentelle Unter suchungen laminarer Grenzschichten bei instabiler Schichtung. Proc. XI Intl. Cong. Appl. Mech., 815.
Wortmann, F. X. (1964b). Experimental investigation of vortex occurrence at transition in unstable boundary-layers. AFOSR Report, 641280.
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Görtler vortices in growing boundary layers: The leading edge receptivity problem, linear growth and the nonlinear breakdown stage

  • Philip Hall (a1)


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